Graph each function. See Examples 6–8 and the Summary of Graphing Techniques box following Example 9. ƒ(x)=-3(x-2)^2+1

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Identify the basic form of the function: The given function \( f(x) = -3(x-2)^2 + 1 \) is a quadratic function in the form \( f(x) = a(x-h)^2 + k \), which represents a parabola.

Determine the vertex of the parabola: The vertex form \( f(x) = a(x-h)^2 + k \) indicates that the vertex is at the point \((h, k)\). Here, \( h = 2 \) and \( k = 1 \), so the vertex is \((2, 1)\).

Analyze the direction of the parabola: The coefficient \( a = -3 \) is negative, which means the parabola opens downwards.

Identify the axis of symmetry: The axis of symmetry for a parabola in vertex form is the vertical line \( x = h \). Therefore, the axis of symmetry is \( x = 2 \).

Determine the effect of the coefficient \( a \): The value \( |a| = 3 \) indicates a vertical stretch by a factor of 3, making the parabola narrower than the standard \( y = x^2 \) parabola.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the general shape and properties of parabolas is essential for graphing quadratic functions.

Vertex Form of a Quadratic

The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex and the direction in which the parabola opens. In the given function, f(x) = -3(x-2)^2 + 1, the vertex is at (2, 1), and the negative coefficient indicates that the parabola opens downwards.

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In the context of the given quadratic function, the expression (x-2) indicates a horizontal shift to the right by 2 units, while the addition of 1 indicates a vertical shift upwards by 1 unit. The coefficient -3 reflects the graph across the x-axis and vertically stretches it, affecting the width of the parabola.

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